In the manual instructions for porting to this phone, at one point it asks, Is the phone currently running Android or Ubuntu?

CyanogenMod is an Android distribution. The instructions will be the same. I have flashed many OnePlus Ones with Ubuntu Touch using the UBPorts installer. In each case they were previously running either the original CyanogenOS shipped by OnePlus or recent builds of LineageOS (the continuation of the now-defunct CyanogenMod project).

Incidentally, I'm guessing you mean "install." "Port" is usually used in these parts to refer to the process of changing the Ubuntu Touch code in order to run on a new device, and UBPorts got UT to run on the OnePlus One long ago.